We study the search process of a target on a rapidly folding polymer ("DNA") by an ensemble of particles ("proteins"), whose search combines 1D diffusion along the chain, Lévy type diffusion mediated by chain looping, and volume exchange. A rich behavior of the search process is obtained with respect to the physical parameters, in particular, for the optimal search.

Intermittent search processes switch between local Brownian search events and ballistic relocation phases. We demonstrate analytically and numerically in one dimension that when relocation times are Lévy distributed, resulting in a Lévy walk dynamics, the search process significantly outperforms the previously investigated case of exponentially distributed relocation times: The resulting Lévy walks reduce oversampling and thus further optimize the intermittent search strategy in the critical situation of rare targets. We also show that a searching agent that uses the Lévy strategy is much less sensitive to the target density, which would require considerably less adaptation by the searcher.random processes | optimization | Lévy walk | movement ecology R andom search processes occur in many areas, from chemical reactions of diffusing reactants (1) to the foraging behavior of bacteria and animals (2, 3). Of general importance is the search efficiency. Brownian search in one and two dimensions involves frequent returns to an area, leading to oversampling. Higher efficiency, can be achieved, for instance, by facilitated diffusion in gene regulation (4) or by controlled motion in foraging (2, 3). From theoretical and data analysis Lévy strategies, in which the searching agent performs excursions whose length is drawn from distributions with a heavy tailfor 0 < α < 2 were shown to be advantageous (5-16); occasional long excursions assist in exploring previously unvisited areas and significantly reduce oversampling.As an alternative to Lévy search, intermittent strategies have been introduced to improve the efficiency of diffusive search (17-21). Intermittent search requires that the searcher occasionally shifts focus from the search and concentrates on fast relocation. The relocation phase implies that the searcher is wasting time in the short run because the target cannot be spotted during it. However, the overall search efficiency is improved by introducing the searcher to previously unexplored areas (17-21).In refs. 18 and 20 relocation events were assumed to occur in a random direction for exponentially distributed time spans, giving rise to a Markovian process. We show here analytically and numerically in one dimension that this is only a partial solution to oversampling, as eventually the central limit theorem (CLT) reduces the process to a Brownian random walk with jumps on the scale of vτ 2 , where τ 2 is the typical time spent in a relocation event. In practice, revisits can be reduced by adjusting the average time spent in search and relocation phases to the density of targets. Lévy strategies, on the other hand, fundamentally circumvent the CLT, and we here demonstrate a twofold advantage of them over the exponential distribution: Lévy walk intermittent processes find the target faster than exponential strategies in the critical case of rare targets, and their performance is much less dependent on adapting to the target density. Intermittent Search with Lévy RelocationsGeneralizing the model from ref. 20, w...

In this study we derive a single-particle equation of motion, from first principles, starting out with a microscopic description of a tracer particle in a one-dimensional many-particle system with a general two-body interaction potential. Using a harmonization technique, we show that the resulting dynamical equation belongs to the class of fractional Langevin equations, a stochastic framework which has been proposed in a large body of works as a means of describing anomalous dynamics. Our work sheds light on the fundamental assumptions of these phenomenological models and a relation derived by Kollmann.

Many genetic processes depend on proteins interacting with specific sequences on DNA. Despite the large excess of nonspecific DNA in the cell, proteins can locate their targets rapidly. After initial nonspecific binding, they are believed to find the target site by 1D diffusion (''sliding'') interspersed by 3D dissociation/reassociation, a process usually referred to as facilitated diffusion. The 3D events combine short intrasegmental ''hops'' along the DNA contour, intersegmental ''jumps'' between nearby DNA segments, and longer volume ''excursions.'' The impact of DNA conformation on the search pathway is, however, still unknown. Here, we show direct evidence that DNA coiling influences the specific association rate of EcoRV restriction enzymes. Using optical tweezers together with a fast buffer exchange system, we obtained association times of EcoRV on single DNA molecules as a function of DNA extension, separating intersegmental jumping from other search pathways. Depending on salt concentration, targeting rates almost double when the DNA conformation is changed from fully extended to a coiled configuration. Quantitative analysis by an extended facilitated diffusion model reveals that only a fraction of enzymes are ready to bind to DNA. Generalizing our results to the crowded environment of the cell we predict a major impact of intersegmental jumps on target localization speed on DNA.A n essential feature in biological processes on DNA is the ability of proteins to quickly locate specific DNA sequences in a vast surplus of nonspecific DNA (1, 2). A protein's search for the target site is thought to be accelerated by facilitated diffusion along nonspecific DNA (3-6). Recent work has yielded considerable insight in the possible search strategies of sitespecific proteins (7-15). Assisted by DNA looping some proteins can, for instance, intermittently bind to two DNA segments simultaneously. This way they can directly move from one to another chemically remote segment (16). This intersegmental transfer accelerates target finding on DNA because it assumes a constantly changing random configuration (11). Here, we demonstrate and quantify a similar mechanism, intersegmental jumping, for proteins with only one DNA-binding site.Little experimental work on facilitated diffusion is available. To date, most studies have been investigating DNA cleavage by restriction enzymes in bulk assays, measuring association times as a function of DNA length, or monitoring processivity on DNA constructs with two sites (7,(17)(18)(19)(20)(21). Although in these biochemical assays valuable information can be obtained, association rates of proteins to specific sites are difficult to measure, and the underlying kinetics of the target search mechanism are often obscured. Furthermore, it remains experimentally challenging to distinguish 1D and 3D search pathways. In previous singlemolecule assays only pure 1D protein search has been addressed (21-24). Here, we present single-molecule measurements of DNA cleavage by EcoRV on individual plasm...